# stationkeeping and transfer trajectory design for spacecraft stationkeeping and transfer trajectory

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STATIONKEEPING AND TRANSFER TRAJECTORY DESIGN FOR SPACECRAFT IN CISLUNAR SPACE

Diane C. Davis,* Sean M. Phillips,† Kathleen C. Howell,‡ Srianish Vutukuri,§ and Brian P. McCarthy§

NASA’s Deep Space Gateway (DSG) will serve as a staging platform for human

missions beyond the Earth-Moon system and a proving ground for inhabited

deep space flight. With a Near Rectilinear Halo Orbit (NRHO) serving as its

primary long-term orbit, the DSG is planned to execute excursions to other

destinations in cislunar space. The current study explores the details of

generating NRHOs in high-fidelity force models. It then investigates the cost of

stationkeeping the primary and destination orbits. Finally, Poincaré maps are

employed in a visual design process for preliminary transfer design between

candidate orbits in cislunar space.

INTRODUCTION

NASA’s recently announced Deep Space Gateway1 (DSG) is planned to take advantage of the complex

multibody dynamical region in cislunar space. Starting with the Exploration series of missions in the mid-

2020s, NASA plans to assemble the DSG in a Near Rectilinear Halo Orbit (NRHO) near the Moon.2 A

member of the halo family of orbits, the candidate NRHO is characterized by a period of approximately

one week, nearly polar passages of the Moon at perilune, and behavior that exhibits nearly stable

characteristics. By the end of the 2020s, NASA plans to exercise the capability to transfer the gateway

spacecraft from the NRHO to other orbits in cislunar space as a demonstration and shakedown of deep

space flight capabilities. Candidate destinations include other members of the L1 and L2 halo families as

well as butterfly orbits, among other options.

These candidate destination orbits exist within periodic orbit families in the dynamical environment

described by the Circular Restricted 3-Body Problem (CR3BP)3 and persist as quasi-periodic trajectories

when transitioned to higher-fidelity models. Multiple methods exist for transitioning an orbit from a

periodic solution in the CR3BP to a quasi-periodic orbit in an ephemeris model;4,5,6 specific

implementations affect the resulting orbit. This study discusses the families of orbits under investigation

and their properties viewed in the CR3BP, and then explores the transition into higher-fidelity models.

A second topic of investigation is stationkeeping of the spacecraft in selected primary and destination

orbits. The orbits under consideration have different stability properties and require different strategies for

long-term orbit maintenance. Multiple strategies for stationkeeping libration point orbits have been

previously investigated; overviews of various methods appear in Folta et al.7 and Guzzetti et al.8 Halo orbit

stationkeeping has been investigated across the L1 and L2 families in the CR3BP by Folta et al.,9,10 and

* Principal Systems Engineer, a.i. solutions, Inc., 2224 Bay Area Blvd, Houston TX 77058, diane.davis@ai-solutions.com. † Principal Software Engineer, a.i. solutions, Inc., 4500 Forbes Blvd., Lanham MD 20706, sean.phillips@ai-solutions.com. ‡‡ Hsu Lo Distinguished Professor, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall of Engineering, 701 W. Stadium Ave., West Lafayette, IN 47907-2045, howell@purdue.edu. Fellow AAS; Fellow AIAA. § Graduate Student, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall of Engineering, 701 W. Stadium

Ave., West Lafayette, IN 47907-2045, svutukur@purdue.edu and mccart71@purdue.edu.

AAS 17-826

mailto:diane.davis@ai-solutions.com mailto:svutukur@purdue.edu

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methods have been proven in flight for specific halos in the Sun-Earth system11 and the Earth-Moon

system.12 Stationkeeping of the L2 southern NRHOs is studied by Guzzetti et al.8 and Davis et al.,13 who

propose low-cost strategies for long-term orbit maintenance of crewed and uncrewed spacecraft. The

current study builds on previous efforts, exploring stationkeeping of the L1 NRHOs as well as looking

broadly at stationkeeping across the L1 and L2 Halo families. Then, stationkeeping of the southern L2

butterfly family is investigated.

A third topic of investigation is the design of transfer trajectories between a starting NRHO and various

destination orbits. Transfers between multibody orbits have been investigated by many researchers.14,15,16

The current study employs periapsis Poincaré maps17,18 for preliminary transfer design. Maps are overlaid

to generate initial transfer arcs that are then corrected into continuous trajectories in the CR3BP. NRHO to

unstable halo orbit transfer orbits are explored, and two families of NRHO to butterfly orbit transfers are

generated.

DYNAMICAL MODELS

In this investigation, two dynamical models are employed. The CR3BP provides the framework for

generation and analysis of libration point orbit families and preliminary transfer design, and an n-body

ephemeris model simulates higher-fidelity mission applications.

The CR3BP describes the motion of a massless spacecraft affected by two primary gravitational bodies

such as the Earth and the Moon. The model assumes that the two primary bodies are point masses orbiting

their center of mass in circular orbits. The spacecraft moves freely under the influence of the two primaries,

and its motion is described relative to a rotating reference frame. No closed-form solution exists to the

CR3BP equations of motion, but five equilibrium solutions, the libration points, are denoted L1 through L5.

Stable and unstable periodic orbit families are found in the vicinity of the libration points. A single integral

of the motion exists in the CR3BP. The Jacobi integral, or Jacobi constant, J, is written

𝐽 = 2𝑈∗ − 𝑣2 (1)

where v is the velocity magnitude of the spacecraft relative to the rotating frame and U* is the pseudo-

potential function.

A higher fidelity model incorporates N-body equations of motion and NAIF planetary ephemerides. A

Moon-centered J2000 inertial coordinate frame is selected for integration, with point-mass perturbations by

the Sun and Earth included. In some simulations, the Moon’s gravity is modeled using the GRAIL

(GRGM660PRIM) model truncated to degree and order 8.

EARTH-MOON HALO AND BUTTERFLY ORBITS

Many families of periodic and quasi-periodic orbits exist in the vicinity of the five libration points in the

CR3BP.19 The halo orbits20 are well-known families near the collinear libration points L1, L2, and L3. In

the Earth-Moon system, the L1 and L2 families bifurcate from planar Lyapunov orbits around the libration

points and evolve out of plane until they approach the Moon. Representative halos appear in Figure 1. A

portion of the L1 halo family appears in Figure 1a. The L1 family extends from the planar Lyapunov orbit

around L1 and grows out of plane until the perilune radii are well within the Moon’s radius. Since orbits

that intersect with the Moon’s surface are not applicable to the current study, only halos with perilune

radius rp > 1,800 km appear in Figure 1a. Note that as perilune radii approach the surface of the Moon, the

apolune radii of the L1 halos grow. This feature is in contrast to the L2 halo orbits, appearing in Figure 1b.

As the L2 halo orbits approach the Moon, their apolune radii decrease. The orbital periods as a function of

rp appear in Figure 2a. The orbital periods of halos in the L2 family range from 14.8 days near the libration

point down to about 5.9 days near the lunar surface. The L1 halos, in contrast, have a maximum orbital

period of about 12 days at an rp of ~43,500 km and a minimum period of about 7.5 days before the apolune

radii begin to increase at rp ~ 4,500 km; the periods of the L1 halos increase as rp values decrease further.

The NRHO21 portions of the L1 and L2 halo families are defined by their stability properties. A stability

index22 is defined as a function of the maximum eigenvalue of the monodromy matrix, i.e., the state

transition matrix (STM) associated with the halo orbit after precisely one revolution. The stability index is

evaluated here as

𝜈 = 1

2 (𝜆𝑚𝑎𝑥 +

1

𝜆𝑚𝑎𝑥 ) (2)

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Figure 1. Periodic Orbits in the CR3BP: The portion of the L1 southern halo family that remain

above the Moon’s surface at perilune (a); the L2 southern halo family (b); L1 and L2 NRHOs (c); and

the L2 southern butterfly family (d).

A halo orbit characterized by a stability index equal to one is considered marginally stable from the

linear analysis. A stability index greater than one corresponds to an unstable halo orbit; the higher the value

of ν, the faster a perturbed halo orbit will tend to depart its nominal path. The stability index for the L1 and

L2 halo families appears as a function of rp in Figure 2a. The unstable, nearly planar halo orbits exist at the

far right of the plot, characterized by high stability indices and large perilune distances. As the halo families

evolve out of plane, their stability indices decrease as the orbits approach the Moon. At v

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